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Let $(Y_t)$ a real-valued stationary Markov chain and $u$ some positive threshold. We assume that for $y>u$, $$Y_{t+1}|\{Y_t=y\}\sim\mathcal{N}(\alpha y+\mu y^\beta,\sigma^2 y^{2\beta})$$

I want to find the maximum likelihood estimator of $(\alpha,\beta,\mu,\sigma)$, but I'm not sure how I should write the likelihood function, since my data are not independent, but they're not a Markov chain either (there are only some clusters of observations above the threshold).

What I did in R:

Yt=Y[1:(length(X)-1)]
Yt1=Y[2:length(X)]

ind=which(Yt>u)

LL=function(alpha,beta,mu,sig){  
  R=dnorm(Yt1[ind],mean=alpha*Yt[ind]+mu*Yt[ind]^beta,sd = sig^2*Yt[ind]^(2*beta))
  -sum(log(R))
}

estim_ml=mle(minuslogl = LL,start = list(alpha=0.7,beta=0.5,mu=0,sig=1))

It works and it gives me estimates that are not absurd, but I think this is not correct as it's just the product of densities, just like independent data. Is there a way to find the likelihood function? If not, is what I did a good approximation?

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If I understand correctly, you have a collection of $Y_t:t=1,2,...T$. Moreover, your model for $Y_t$ specifies that it is conditional on $Y_{t-1}$. What you can do is a recursive scheme that is quite common in dynamic time series analysis. You essentially condition observation for observation as follows:

$f(Y_t|\alpha, \beta, \mu, \sigma; \{Y_k\}_{k=1}^{t-1}) = f(Y_t|\alpha, \beta, \mu, \sigma; Y_{t-1})\cdot f(Y_{t-1}|\alpha, \beta, \mu, \sigma; \{Y_k\}_{k=1}^{t-2}) = \dots = \Pi_{t=2}^T f(Y_t|\alpha, \beta, \mu, \sigma; Y_{t-1}):= F(Y)$

The recursion happens at $\dots$, where we extend the conditioning principle to the next $T-1$ observations. Notice that the final product runs from $2$ to $T$, because we have to take $Y_1$ as given (aka, as 'initial condition') for this approach to work. It will appear in the conditional pdf of $Y_2$, but its pdf will not be part of the expression to be maximized. Once F(Y) is obtained, you can apply standard MLE techniques to it.

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  • $\begingroup$ Thank you for your answer. I understand what you mean, but how do you account for the fact that I know the distribution of $Y_{t}$ given $Y_{t-1}$ only when $Y_{t-1}>u$? If $\{t_1,\dots,t_n\}$ denotes the set of times where $Y_t>u$, the observations that I need to consider are $y_{t_1},y_{t_1+1},y_{t_2},y_{t_2+1},\dots,y_{t_n},y_{t_n+1}$. $\endgroup$
    – Augustin
    Commented Jan 20, 2016 at 12:55
  • $\begingroup$ Are you interested in the distribution of $Y_t|Y_{t-1}$ or of $Y_t|Y_{t-1}, Y_{t-1}>u$? If it's the latter you are interested in, you can still apply the technique for every block of length $k$ for which the first $k-1$ observations are larger than $u$ (call the result $F_j(Y)$) and maximize the product $\Pi_{j=1}^{K}F_j(Y)$ over the parameters of interest. If you are interested in $Y_t|Y_{t-1}$, I will have to look into censored regression techniques again. It's been a while ;) $\endgroup$
    – Jeremias K
    Commented Jan 20, 2016 at 13:18

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